Vargas noted that Cornyn voted against the DREAM Act, which would have created a pathway to citizenship for immigrants brought to the country illegally when they were children.

Vargas, a law school graduate who was illegally brought to the U.S. when he was 5 years old, argued that the latest Gang of Eight immigration proposal "addresses many, if not all, of Cornyn's demands for border security."

But he claimed that Cornyn will likely oppose the proposal out of fear of a Tea Party primary challenge.

Cornyn is a member of the Senate Judiciary Committee, which will consider the legislation.

"Unlike previous attempts, Cornyn's effort to doom immigration reform will not have the same potency or success rate," Vargas wrote.

"Cornyn can still take advantage of this last chance to prove to Texans a legacy of firmness in leadership to take on immigration reform that includes not only border security but a practical system that unites all families," he added.

At a hearing earlier this week, Cornyn said it is critical for Congress to begin examining immigration reform and that the proposal makes a "number of positive improvements to our immigration system."

But he said the border security provisions "fall short" and that "without major changes, the bill could do more harm than good."

"The bill is a little over a week old, and Sen. Cornyn has repeatedly praised his colleagues’ efforts," Drew Brandewie, a spokesman for Cornyn, said in an email. "Last week, Sen. Cornyn laid out areas where he agreed and disagreed with the Gang of 8. He looks forward to the mark up and the continued debate on this important, complex issue."